Introduction to quantitative methods

Objectives When you have completed this study unit you will be able to:  identify some different number systems  round-up numbers and correct them to significant figures  carry out ca

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Business Management

INTRODUCTION TO QUANTITATIVE

METHODS

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INTRODUCTION TO QUANTITATIVE METHODS

Contents

4 Simultaneous, Quadratic, Exponential and Logarithmic Equations 87

6 Introduction to Statistics and Data Analysis 171

(Continued over)

G Percentages 27

The skill of being able to handle numbers well is very important in many jobs You may often

be called upon to handle numbers and express yourself clearly in numerical form – and this

is part of effective communication, as well as being essential to understanding and solvingmany business problems

Sometimes we can express numerical information more clearly in pictures – for example, incharts and diagrams We shall consider this later in the course Firstly though, we need toexamine the basic processes of manipulating numbers themselves In this study unit, weshall start by reviewing a number of basic numerical concepts and operations It is likely thatyou will be familiar with at least some of these, but it is essential to ensure that you fullyunderstand the basics before moving on to some of the more advanced applications in laterunits

Throughout the course, as this is a practical subject, there will be plenty of practice questionsfor you to work through To get the most out of the material, we recommend strongly that you

do attempt all these questions

One initial word about calculators It is perfectly acceptable to use calculators to performvirtually all arithmetic operations – indeed, it is accepted practice in examinations You shouldtherefore get to know how to use one quickly and accurately However, you should alsoensure that you know exactly how to perform all the same operations manually

Understanding how the principles work is essential if you are to use the calculator correctly

Objectives

When you have completed this study unit you will be able to:

 identify some different number systems

 round-up numbers and correct them to significant figures

 carry out calculations involving the processes of addition, subtraction, multiplicationand division

 manipulate negative numbers

 cancel fractions down to their simplest terms

 change improper fractions to mixed numbers or integers, and vice versa

 add, subtract, multiply and divide fractions and mixed numbers

 carry out calculations using decimals

 calculate ratios and percentages, carry out calculations using percentages and divide agiven quantity according to a ratio

 explain the terms index, power, root, reciprocal and factorial

A NUMBER SYSTEMS

Denary Number System

We are going to begin by looking at the number system we use every day

Because ten figures (i.e symbols 0 to 9) are used, we call this number system the base ten system or denary system This is illustrated in the following table.

Seventh

column

Sixthcolumn

Fifthcolumn

Fourthcolumn

Thirdcolumn

Secondcolumn

Firstcolumn

Millions

Hundredsofthousands

Tens ofthousands Thousands Hundreds Tens Units

Figure 1.1: The denary system

The number of columns indicates the actual amounts involved Consider the number 289.This means that, in this number, there are:

positive We indicate a negative number by a minus sign () A common example of the use

of negative numbers is in the measurement of temperature – for example,20°C (i.e 20°Cbelow zero)

We could indicate a positive number by a plus sign (), but in practice this is not necessaryand we adopt the convention that, say, 73 means73

Roman Number System

We have seen that position is very important in the number system that we normally use.This is equally true of the Roman numerical system, but this system uses letters instead offigures Although we invariably use the normal denary system for calculations, Roman

numerals are still in occasional use in, for example, the dates on film productions, tabulation,house numbers, etc

Here are some Roman numbers and their denary equivalent:

Binary System

As we have said, our numbering system is a base ten system By contrast, computers use

base two, or the binary system This uses just two symbols (figures), 0 and 1.

In this system, counting from 0 to 1 uses up all the symbols, so a new column has to bestarted when counting to 2, and similarly the third column at 4

Figure 1.2: The binary system

This system is necessary because computers can only easily recognise two figures – zero orone, corresponding to "components" of the computer being either switched off or on When anumber is transferred to the computer, it is translated into the binary system

B NUMBERS – APPROXIMATION AND INTEGERS

Approximation

Every figure used in a particular number has a meaning However, on some occasions,precision is not needed and may even hinder communication For example, if you are

calculating the cost of the journey, you need only express the distance to the nearest mile

It is important to understand the methods of approximation used in number language

(a) Rounding off

Suppose you are given the figure 27,836 and asked to quote it to the nearest thousand.Clearly, the figure lies between 27,000 and 28,000 The halfway point between the two

is 27,500 The number 27,836 is more than this and so, to the nearest thousand, theanswer would be 28,000

There are strict mathematical rules for deciding whether to round up or down to thenearest figure:

Look at the digit after the one in the last required column Then:

Thus, when rounding off 236 to the nearest hundred, the digit after the one in the lastrequired column is 3 (the last required column being the hundreds column) and theanswer would be to round down to 200 When rounding off 3,455 to the nearest ten,the last required column is the tens column and the digit after that is 5, so we wouldround up to 3,460

(However, you should note that different rules may apply in certain situations – forexample, in VAT calculations, the Customs and Excise rule is always to round down tothe nearest one pence.)

(b) Significant figures

By significant figures we mean all figures other than zeros at the beginning or end of anumber Thus:

 632,000 has three significant figures as the zeros are not significant

 000,632 also has only three significant figures

 630,002 has six significant figures – the zeros here are significant as they occur

in the middle of the number

We can round off a number to a specified number of significant figures This is likely to

be done to aid the process of communication – i.e to make the number languageeasier to understand – and is particularly important when we consider decimals later inthe unit

Consider the figure 2,017 This has four significant figures To write it correct to threesignificant figures, we need to drop the fourth figure – i.e the 7 We do this by using therounding off rule – round down from 4 or round up from 5 – so 2,017 correct to threesignificant figures becomes 2,020 To write 2,017 correct to one significant figure, weconsider the second significant figure, the zero, and apply the rule for rounding off,giving 2,000 Do not forget to include the insignificant zeros, so that the digits keep

their correct place values.

Integers

Integers are simply whole numbers Thus, 0, 1, 2, 3, 4, etc, are integers, as are all the

negative whole numbers

The following are not integers:

¾, 1½, 7¼, 0.45, 1.8, 20.25

By adjusting any number to the nearest whole number, we obtain an integer Thus, a number

which is not an integer can be approximated to become one by rounding off.

Questions for Practice 1

1 Round off the following the figures to the nearest thousand:

3 Correct the following to three significant figures:

When carrying out arithmetic operations, remember that the position of each digit in a

number is important, so layout and neatness matter

The figures in the right hand column (the units column) are added first, then those in the nextcolumn to the left and so on Where the sum of the numbers in a column amount to a figuregreater than nine (for example, in the above calculation, the sum of the units column comes

to 17), only the last number is inserted into the total and the first number is carried forward to

be added into the next column Here, it is 1 or, more precisely, one ten which is carried

forward into the "tens" column

Where you have several columns of numbers to add (as may be the case with certain

statistical or financial information), the columns can be cross-checked to produce checktotals which verify the accuracy of the addition:

As with addition, you must work column by column, starting from the right – the units column.Now consider this calculation:

34

18Total 16

Where it is not possible to subtract one number from another (as in the units column whereyou cannot take 8 from 4), the procedure is to "borrow" one from the next column to the leftand add it to the first number (the 4) The one that you borrow is, in fact, one ten – so addingthat to the first number makes 14, and subtracting 8 from that allows you to put 6 into thetotal Moving on to the next column, it is essential to put back the one borrowed and this isadded to the number to be subtracted Thus, the second column now becomes 3 (1  1), or

In practice, it is likely that you would use a calculator for this type of "long multiplication", but

we shall briefly review the principles of doing it manually

Before doing so though, we shall note the names given to the various elements involved in amultiplication operation:

 the multiplicand is the number to be multiplied – in this example, it is 246

 the multiplier is the number by which the multiplicand is to be multiplied – here it is 23

and

 the answer to a multiplication operation is called the product – so to find the product of

two or more numbers, you multiply them together

Returning to our example of long multiplication, the first operation is to multiply the

multiplicand by the last number of the multiplier (the number of units, which is 3) and enterthe product as a subtotal – this shown here as 738 Then we multiply the multiplicand by thenext number of the multiplier (the number of tens, which is 2) However, here we have toremember that we are multiplying by tens and so we need to add a zero into the units column

of this subtotal before entering the product for this operation (492) Thus, the actual product

of 246 20 is 4,920 Finally, the subtotals are added to give the product for the whole

operation:

246

 23Subtotal (units) 738

Just as there were three names for the various elements involved in a multiplication

operation, so there are three names or terms used in division:

 the dividend is the number to be divided – in this example, it is 360

 the divisor is the number by which the dividend is to be divided – here it is 24 and

 the answer to a division operation is called the quotient – here the quotient for our

example is 15

Division involving a single number divisor is relatively easy – for example, 16 2  8 Forlarger numbers, the task becomes more complicated and you should use a calculator for alllong division

The Rule of Priority

Calculations are often more complicated than a simple list of numbers to be added and/orsubtracted They may involve a number of different operations – for example:

8 6  3 

28

When faced with such a calculation, the rule of priority in arithmetical operations requiresthat:

division and multiplication are carried out before addition and subtraction.

Thus, in the above example, the procedure would be:

(a) first the multiplication and division: 8 6  48;

2

8  4(b) then, putting the results of the first step into the calculation, we can do the addition andsubtraction: 48 3  4  49

Brackets

Brackets are used extensively in arithmetic, and we shall meet them again when we explorealgebra

Look at the previous calculation again It is not really clear whether we should:

(a) multiply the eight by six; or

(b) multiply the eight by "six minus three" (i.e by three); or even

(c) multiply the eight by "six minus three plus four" (i.e by seven)

We use brackets to separate particular elements in the calculation so that we know exactlywhich operations to perform on which numbers Thus, we could have written the abovecalculation as follows:

(8 6)  3 

28

This makes it perfectly clear what numbers are to be multiplied

Consider another example:

Note that we can put brackets within brackets to further separate elements within the

calculation and clarify how it is to be performed

The rule of priority explained above now needs to be slightly modified It must be strictly

followed except when brackets are included In that case, the contents of the brackets are

evaluated first If there are brackets within brackets, then the innermost brackets are

Thus we can see that brackets are important in signifying priority as well as showing how totreat particular operations.

Questions for Practice 2

1 Work out the answers to the following, using a calculator if necessary

Now check your answers with those given at the end of the unit.

D DEALING WITH NEGATIVE NUMBERS

Addition and Subtraction

We can use a number scale, like the markings on a ruler, as a simple picture of addition andsubtraction Thus, you can visualise the addition of, say, three and four by finding the position

of three on the scale and counting off a further four divisions to the right, as follows:

This brings us to the expected answer of seven (3 4)

Similarly we can view subtraction as moving to the left on the scale Thus, 6 minus 4 is

represented as follows:

This illustrates the sum 6 4  2

Negative numbers can also be represented on the number scale All we do is extend thescale to the left of zero:

We can now visualise the addition and subtraction of negative numbers in the same way asabove Consider the sum (3)  4 Start at 3 on the scale and count four divisions to theright:

4 (3)  4  3  7

To sum up:

Consider the following examples (Draw your own number scales to help visualise the sums

if you find it helpful.)

7 (3)  7  3  10

2 (6)  2  6  4

Multiplication and Division

The rules for multiplication and division are as follows:

Type of calculation Product/Quotient Example

Minus times plus minus 3  7  21Minus times minus plus 7  (3)  21Minus divide by plus minus 21  3  7Minus divide by minus plus 21  (3)  7Plus divide by minus minus 21 (3)  7

Figure 1.3: Multiplying and dividing using negative numbers

Questions for Practice 3

Solve the following sums, using a number scale if necessary:

A fraction is a part of whole number

When two or more whole numbers are multiplied together, the product is always anotherwhole number In contrast, the division of one whole number by another does not alwaysresult in a whole number Consider the following two examples:

12 5  2, with a remainder of 2

10 3  3, with a remainder of 1

The remainder is not a whole number, but a part of a whole number – so, in each case, theresult of the division is a whole number and a fraction of a whole number The fraction isexpressed as the remainder divided by the original divisor and the way in which it is shown,then, is in the same form as a division term:

12 5  2, with a remainder of 2 parts of 5, or two divided by five, or

52

10 3  3, with a remainder of 1 part of 3, or one divided by three, or

31

Each part of the fraction has a specific terminology:

 the top number is the numerator and

 the bottom number is the denominator.

Thus, for the above fractions, the numerators are 2 and 1, and the denominators are 5 and 3.When two numbers are divided, there are three possible outcomes

(a) The numerator is larger than the denominator

In this case, the fraction is known as an improper fraction The result of dividing out an

improper fraction is a whole number plus a part of one whole, as in the two examplesabove Taking two more examples:

9

42  4 whole parts and a remainder of 6 parts of 9, or 4

96

24

30  1 whole part and a remainder of 6 parts of 24, or 1

246

(b) The denominator is larger than the numerator

In this case, the fraction is known as a proper fraction The result of dividing out a

proper fraction is only a part of one whole and therefore a proper fraction has a value ofless than one

Examples of proper fractions are:

5

1

,3

2,7

4,5

4, etc

(c) The numerator is the same as the denominator

In this case, the expression is not a true fraction The result of dividing out is one wholepart and no remainder The answer must be 1 For example:

5

5  5  5  1

Fractions are often thought of as being quite difficult However, they are not really hard aslong as you learn the basic rules about how they can be manipulated We shall start bylooking at these rules before going on to examine their application in performing arithmeticoperations on fractions

Basic Rules for Fractions

 Cancelling down

Both the numerator and the denominator in a fraction can be any number The

following are all valid fractions:

5

2

,50

20,93

47,96

The rule in expressing fractions, though, is always to reduce the fraction to the smallest

possible whole numbers for both its numerator and denominator The resulting fraction

is then in its lowest possible terms.

The process of reducing a fraction to its lowest possible terms is called cancelling

down To do this, we follow another simple rule Divide both the numerator and

denominator by the largest whole number which will divide into them exactly

Consider the following fraction from those listed previously:

50

20

Which whole number will divide into both 20 and 50? There are several: 2, 5, and 10.However the rule states that we use the largest whole number, so we should use 10.Dividing both parts of the fraction by 10 reduces it to:

5

2

Note that we have not changed the value of the fraction, just the way in which the parts

of the whole are expressed Thus:

50

20 

52

A fraction which has not been reduced to its lowest possible terms is called a vulgar

3231

3165



 7755

This gives us a new form of the same fraction, and we can now see that both terms can

be further divided – this time by 11:

1177

1155



75

So, large vulgar fractions can often be reduced in stages to their lowest possible terms

 Changing the denominator to required number

The previous process works in reverse If we wanted to express a given fraction in adifferent way, we can do that by multiplying both the numerator and denominator by thesame number So, for example,

by multiplying both terms by 4

If we wanted to change the denominator to a specific number – for example, to express

½ in eighths – the rule is to divide the required denominator by the existing

denominator (8 2  4) and then multiply both terms of the original fraction by thisresult:

42

41



 84

Note again that we have not changed the value of the fraction, just the way in which theparts of the whole are expressed

 Changing an improper fraction into a whole or mixed number

Suppose we are given the following improper fraction:

8

35

The rule for converting this is to divide the numerator by the denominator and placeany remainder over the denominator Thus we find out how many whole parts there areand how many parts of the whole (in this case, eighths) are left over:

8

35  35  8  4 and a remainder of three  4

83

This result is known as a mixed number – one comprising a whole number and a

fraction

 Changing a mixed number into an improper fraction

This is the exact reverse of the above operation The rule is to multiply the whole

number by the denominator in the fraction, then add the numerator of the fraction tothis product and place the sum over the denominator

For example, to write 3

Questions for Practice 4

1 Reduce the following fractions to their lowest possible terms:

(a)

36

9

(b)648

(c)

70

21

(d)5418

(e)

49

14

(f)15025

(g)

81

36

(h)13545

(i)

192

24

(j)8833

2 Change the following fractions as indicated:

(c)

6

25

(d)331

4 Change the following mixed numbers to improper fractions:

Now check your answers with those given at the end of the unit.

Adding and Subtracting with Fractions

(a) Proper fractions

Adding or subtracting fractions depends upon them having the same denominator It isnot possible to do these operations if the denominators are different

Where the denominators are the same, adding or subtracting is just a simple case ofapplying the operation to numerators For example:

7

3 

7

2 75

8

5 

81 8

4 21

Where the denominators are not the same, we need to change the form of expression

so that they become the same This process is called finding the common denominator.Consider the following calculation:

5

2 

103

We cannot add these two fractions together as they stand – we have to find the

common denominator In this case, we can change the expression of the first fraction

by multiplying both the numerator and denominator by 2, thus making its denominator

10 – the same as the second fraction:

104 

103 107

Not all sums are as simple as this It is often the case that we have to change the form

of expression of all the fractions in a sum in order to give them all the same

The lowest common denominator for these three fractions is 20 We must then convertall three to the new denominator by applying the rule explained above for changing afraction to a required denominator:

12 

205 

202 20

15 43

The rule is simple:

 find the lowest common denominator – and where this is not readily apparent, acommon denominator may be found by multiplying the denominators together

 change all the fractions in the calculation to this common denominator

 then add (or subtract) the numerators and place the result over the commondenominator

(b) Mixed numbers

When adding mixed numbers, we deal with the integers and the fractions separately.The procedure is as follows:

 first add all the integers together

 then add the fractions together as explained above, and then

 add together the sum of the integers and the fractions

Consider the following calculation:

3 4

4

1  5

21  243

141

241

 14 

46

 14  1

21

 15

21

In general, the same principles are applied to subtracting mixed numbers For example:

16

36

5

 3 6

2  331

However, the procedure is slightly different where the second fraction is larger than thefirst Consider this calculation:

4

21  1

65

Here, it is not possible to subtract the fractions even after changing them to the

common denominator:

6

3 

65

To complete the calculation we need to take one whole number from the integer in thefirst mixed number, convert it to a fraction with the same common denominator and add

it to the fraction in the mixed number Thus:

36

69

We can now carry out the subtraction as above:

4

21  1

6

5  36

9  16

56

9

 2 6

4  232

Questions for Practice 5

Carry out the following calculations:

3  121

15

32  3  1

2

1  25

1

16 1

4

3 32

19 2

8

1  2

31  26

1

20 4

4

3  13

2  221

Now check your answers with those given at the end of the unit.

Multiplying and Dividing Fractions

(a) Multiplying a fraction by a whole number

In this case, we simply multiply the numerator by the whole number and place the product (which is now the new numerator) over the old denominator For example:

We often see the word "of" in a calculation involving fractions – for example, ¼ of

£12,000 This is just another way of expressing multiplication

(b) Dividing a fraction by a whole number

In this case, we simply multiply the denominator by the whole number and place the old

numerator over the product (which is now the new denominator) For example:

4

3  5 

54

2

  15

2

(c) Multiplying one fraction by another

In this case, we multiply the numerators by each other and the denominators by eachother, and then reduce the result to the lowest possible terms For example:

23



 

126 21

33



 409

16  5

31

61 

31 6

1 1

3 6

3 21

(e) Dealing with mixed numbers

All that we have said so far in this section applies to the multiplication and division ofproper and improper fractions Before we can apply the same methods to mixed

numbers, we must complete one further step – convert the mixed number to an

improper fraction For example:

5

21 

114 2

11 

114 22

19 1

2 4

38  9

21

(f) Cancelling during multiplication

We have seen how it is possible to reduce fractions to their lowest possible terms bycancelling down This involves dividing both the numerator and the denominator by thesame number, for example:

by dividing both terms by 2

We often show the process of cancelling down during a calculation by crossing out thecancelled terms and inserting the reduced terms as follows:

312

28

32

(It is always helpful to set out all the workings throughout a calculation Some

calculations can be very long, involving many steps If you make a mistake somewhere,

it is much easier to identify the error if all the workings are shown It is also the case

that, in an examination, examiners will give "marks for workings", even if you make amistake in the arithmetic However, if you do not show all the workings, they will not beable to follow the process and see that the error is purely mathematical, rather than inthe steps you have gone through.)

Cancellation can also be used in the multiplication of fractions In this case it can beapplied across the multiplication sign – dividing the numerator of one fraction and thedenominator of the other by the same number For example:

4

1

3 

412

1 161

Remember that every cancelling step must involve a numerator and a denominator.

Thus, the following operation would be wrong:

14

3

312

1

33

Note, too, that you cannot cancel across addition and subtraction signs.

However, you can use cancellation during a division calculation at the step when,

having turned the divisor upside down, you are multiplying the two fractions So,

repeating the calculation involving mixed numbers from above, we could show it asfollows:

Questions for Practice 6

Carry out the following calculations:

Decimals are an alternative way of expressing a particular part of a whole The term

"decimal" means "in relation to ten", and decimals are effectively fractions expressed intenths, hundredths, thousandths, etc

Unlike fractions though, decimals are written completely differently and this makes carryingout arithmetic operations on them much easier

Decimals do not have a numerator or a denominator Rather, the part of the whole is shown

by a number following a decimal point:

 the first number following the decimal point represents the number of tenths, so 0.1 isone-tenth part of a whole and 0.3 is three-tenths of a whole

 the next number to the right, if there is one, represents the number of hundredths, so0.02 is two-hundredths of a whole and 0.67 is six-tenths and seven-hundredths (i.e 67hundredths) of a whole

 the next number to the right, again if there is one, represents the number of

thousandths, so 0.005 is five-thousandths of a whole and 0.134 is 134 thousandths of awhole

and so on up to as many figures as are necessary

We can see the relationship between decimals and fractions as follows:

106

0.01

100

1, 0.67

100

67, 3.81 3

10081

0.001

1000

1, 0.054

1000

54, 5.382 5

1000382

We can see that any fraction with a denominator of ten, one hundred, one thousand, tenthousand, etc can be easily converted into a decimal However, fractions with a differentdenominator are also relatively easy to convert by simple division

Arithmetic with Decimals

Performing addition, subtraction, multiplication and division with decimals is exactly the same

as carrying out those operations on whole numbers The one key point to bear in mind is to

get the position of the decimal point in the right place Layout is all important in achieving

this

(a) Adding and subtracting

To add or subtract decimals, arrange the numbers in columns with the decimal pointsall being in the same vertical line Then you can carry out the arithmetic in the sameway as for whole numbers

For example, add the following: 3.6 7.84  9.172

The layout for this is as follows:

3.67.849.172Total 20.612

Similarly, subtraction is easy provided that you follow the same rule of ensuring that thedecimal points are aligned vertically For example, subtract 18.857 from 63.7:

63.7

18.857Total 44.843

(b) Multiplying decimals

It is easier to use a calculator to multiply decimals, although you should ensure that youtake care to get the decimal point in the correct place when both inputting the figuresand when reading the answer

The principles of multiplying decimals manually are the same as those consideredearlier in the study unit for whole numbers Again, it is important to get the decimalpoint in the right place

There is a simple rule for this – the answer should have the same number of decimalplaces as there were in the question

Consider the calculation of 40 0.5 The simplest way of carrying this out is to ignorethe decimal points in the first instance, multiply the numbers out as if the figures werewhole numbers, and then put back the decimal point in the answer The trick is to getthe position of the decimal point right!

So, ignoring the decimal points, we can multiply the numbers as follows:

If the sum had been 0.4 0.5, what would the answer be?

The multiplication would be the same, but this time there are 2 figures after decimalpoints in the original sum So, we insert the decimal point two places to the right in theanswer:

(c) Division with decimals

Again, it is far easier to use a calculator when dividing with decimals

However, when dividing by ten or by one hundred (or even by a thousand or a million),

we can use the technique noted above in respect of multiplication to find the answerquickly without using a calculator:

 to divide by ten, simply move the decimal point one place to the left – for

example, 68.32 10  6.832

 to divide by one hundred, simply move the decimal point two places to the left –

for example, 68.32 100  0.6832

(d) Converting fractions to decimals, and vice versa

Fractions other than tenths, hundredths, thousandths, etc can also be converted intodecimals It is simply a process of dividing the numerator by the denominator andexpressing the answer in decimal form You can do this manually, but in most

instances it is easier to use a calculator

For example, to change

5

3into decimal form, you divide 3 by 5 (i.e 3 5)

Using a calculator the answer is easily found: 0.6

However, not all fractions will convert so easily into decimals in this way

Consider the fraction

3

1

Dividing this out (or using a calculator) gives the answer 0.3333333 or however many3s for which there is room on the paper or the calculator screen This type of decimal,

where one or more numbers repeat infinitely, is known as a recurring decimal.

In most business applications, we do not need such a degree of precision, so weabbreviate the decimal to a certain size – as we shall discuss in the next section

To change a decimal into a fraction, we simply place the decimal over ten or one

hundred etc, depending upon the size of the decimal Thus, the decimal becomes thenumerator and the denominator is 10 or 100 etc For example:

0.36

10036 

259

3.736 3

1000736  3

12592

Limiting the Number of Decimal Places

We noted that certain fractions do not convert into exact decimals The most common

examples are one-third and two-thirds, which convert into 0.3333333 and 0.6666666 respectively (When referring to these types of recurring decimals, we would normally justquote the decimal as, say, "0.3 recurring".) However, there are also plenty of other fractionswhich do convert into exact decimals, but only at the level of thousandths or greater Forexample:

7

2  0.2857142, or

1711  0.6470568

This gives us a problem: how much detail do we want to know? It is very rare that, in

business applications, we would want to know an answer precise to the millionth part of awhole, although in engineering, that level of precision may be critical Such large decimalsalso present problems in carrying out arithmetic procedures – for example, it is not easy toadd the above two decimals together and multiplying them does not bear thinking about

In most business circumstances then, we limit the number of decimal places to a

manageable number This is usually set at two or three decimal places, so we are only

interested in the level of detail down to hundredths or thousandths We then speak of a

number or an answer as being "correct to two (or three) decimal places".

For decimals with a larger number of decimal places than the required number, we use theprinciples of rounding discussed earlier in the study unit to reduce them to the correct size

Thus, we look at the digit in the decimal after the one in the last required place and then:

 for figures of five or more, round up the figure in the last required place

 for figures of four or less, round down by leaving the figure in the last required placeunchanged

So, if we want to express the decimal form of two-sevenths (0.2857142) correct to threedecimal places, we look at the digit in the fourth decimal place: 7 Following the rule forrounding, we would then round up the number in the third decimal place to give 0.286

Now consider the decimal forms of one-third and two-thirds If we were to express thesecorrect to two decimal places, these would be 0.33 and 0.67 respectively

Working to only two or three decimal places greatly simplifies decimals, but you should notethat the level of accuracy suffers For example, multiplying 0.33 by two gives us 0.66, but wehave shown two-thirds as being 0.67 These types of small error can increase when adding,subtracting, multiplying or dividing several decimals which have been rounded – somethingwhich we shall see later in the course in respect of a set of figures which should add up to

100, but do not In such cases, we need to draw attention to the problem by stating that

"errors are due to rounding"

Questions for Practice 7

1 Carry out the following calculations:

to 3 dp

 4% means four-hundredths of a whole, or four parts of a hundred

1004

 28% means twenty-eight-hundredths of a whole, or twenty-eight parts of a hundred100

important that you are fully conversant with working with them

Arithmetic with Percentages

Multiplication and division have no meaning when applied to percentages, so we need only

be concerned with their addition and subtraction These processes are quite straightforward,but you need to remember that we can only add or subtract percentages if they are parts ofthe same whole

Take an example: if Peter spends 40% of his income on rent and 25% on household

expenses, what percentage of his income remains? Here both percentages are parts of thesame whole – Peter's income – so we can calculate the percentage remaining as follows:100% (40%  25%)  35%

However, consider this example: if Alan eats 20% of his cake and 10% of his pie, what

percentage remains? Here the two percentages are not parts of the same whole – one refers

to cake and the other to pie Therefore, it is not possible to add the two together

Percentages and Fractions

Since a percentage is a part of one hundred, to change a percentage to a fraction you simplydivide it by 100 In effect all this means is that you place the percentage over 100 and thenreduce it to its lowest possible terms

For example:

60%

10060 

52

24%

10024 

256

Changing a fraction into a percentage is the reverse of this process – simply multiply by 100.For example:

Percentages and Decimals

To change a percentage to a decimal, again you divide it by 100 As we saw in the last

section, this is easy – you simply move the decimal point two places to the left Thus:

To find a particular percentage of a given number, change the percentage rate into a fraction

or a decimal and then multiply the given number by that fraction/decimal

For example, to find 15% of £260:

For example, what percentage of cars have been sold if 27 have gone from a total stock of300?

30027  100 

300

2700  9%

Do not forget that percentages refer to parts of the same whole and therefore can only becalculated where the units of measurement of the quantities concerned are the same.

For example, to show 23 pence as a percentage of £2, we need to express both quantities inthe same units before working out the percentage This could be both as pence or both aspounds:

20023  100  11.5%

or

2

23

0

 100  11.5%

There are occasions when we deal with percentages greater than 100%

For example, if a woman sells a car for £1,200 and says that she made 20% profit, howmuch did she buy it for?

The cost plus the profit 120% of cost  £1,200

The cost, therefore, is 100%

To work out the cost, we need to multiply the selling price by

120

100:

Cost 1,200 

120

100  £1,000

To prove that this is correct, we work the calculation through the other way The cost was

£1,000 and she sold it at 20% profit, so the selling price is:

£1,000 20% of £1,000  £(1,000  0.2  1,000) = £1,200, as given above

Questions for Practice 8

1 Change the following fractions into percentages:

(a)

100

3

(b)83

2 Change the following percentages into fractions and reduce them to their lowest

4 Calculate the following:

(a) 60 as a percentage of 300 (b) 25 as a percentage of 75

(c) £25 as a percentage of £1,000 (d) 30 as a percentage of 20

Now check your answers with those given at the end of the unit.

H RATIOS

A ratio is a way of expressing the relationship between two quantities It is essential that thetwo quantities are expressed in the same units of measurement – pence, number of people,etc – or the comparison will not be valid

For example, if we wanted to give the ratio of the width of a table to its depth where the width

is 2 m and the depth is 87 cm, we would have to convert the two quantities to the same unitsbefore giving the ratio Expressing both in centimetres would give the ratio of 200 to 87.Note that the ratio itself is not in any particular unit – it just shows the relationship betweenquantities of the same unit

We use a special symbol (the colon symbol : ) in expressing ratios, so the correct form ofshowing the above ratio would be:

200: 87 (pronounced "200 to 87")

There are two general rules in respect of ratios:

To do this, express the ratio as a fraction by putting the first figure over the second andcancel the resulting fraction Then re-express the ratio in the form of numerator :

The ratio of 85 to 17 is, therefore, 5 : 1

fractions or decimals in them

Consider the ratio of average miles per gallon for cars with petrol engines to that forcars with diesel engines, where the respective figures are 33½ mpg and 47 mpg Wewould not express this as 33½ : 47, but convert the figures to whole numbers by, here,multiplying by 2 to give: 67 : 94

Ratios are used extensively in business to provide information about the way in which oneelement relates to another For example, a common way of analysing business performance

is to compare profits with sales If we look at the ratio of profits to sales across two years, wemay be able to see if this aspect of business performance is improving, staying the same ordeteriorating (This will be examined elsewhere in your studies.)

Another application is to work out quantities according to a particular ratio For example,ratios are often used to express the relationship between partners in a partnership – sharingprofits on a basis of, say, 50 : 50 or 80 : 20

Consider the case of three partners – Ansell, Boddington and Devenish – who share theprofits of their partnership in the ratio of 3 : 1 : 5 If the profit for a year is £18,000, how muchdoes each partner receive?

To divide a quantity according to a given ratio:

 first add the terms of the ratio to find the total number of parts

 then find what fraction each term of the ratio is to the whole, and

 finally divide the total quantity into parts according to the fractions

For the Ansell, Boddington and Devenish partnership, this would be calculated as follows:

The profit is divided into 3 1  5  9 parts.

Questions for Practice 9

1 Express the following sales and profits figures as profit to sales ratios for each firm:

Firm Sales Profits

2 Calculate the distribution of profits for the following partnerships:

(a) X, Y and Z share profits in the ratio of 3 : 4: 5 Total profit is £60,000

(b) F, G and H share profits in the ratio of 2 : 3 : 3 Total profit is £7,200

Now check your answers with those given at the end of the unit.

I FURTHER KEY CONCEPTS

In this last section we shall briefly introduce a number of concepts which will be used

extensively later in the course Here we concentrate mostly on what these concepts mean,and you will get the opportunity to practice their use in later study units However, we doinclude practice in relation to the expression of numbers in standard form

Indices

Indices are found when we multiply a number by itself one or more times The number oftimes that the multiplication is repeated is indicated by a superscript number to the right ofthe number being multiplied Thus:

2 2 is written as 22(the little "2" raised up is the superscript number)

2 2  2 is written as 23

2 2  2  2  2  2  2 is written as 27etc

The number being multiplied (here, 2) is termed the base and the superscript number is called the exponent or index of its power Thus in the case of 26we refer to the term as "tworaised to the sixth power" or "two to the power of six" (Strictly, 22should be referred to as

"two to the power of two", but we normally call it "2 squared"; similarly, 23is referred to as "2cubed".)

The power of a number is the result of multiplying that number a specified number of times.

Thus, 8 is the third power of 2 (23) and 81 is the fourth power of 3 (34)

It is quite possible to have a negative exponent, as in the case of 23 A negative exponent

indicates a reciprocal The reciprocal of any number is one divided by that number For

example:

the reciprocal 5 is

5

1and

23represents 3

2

1

 8

1  0.125

Indices also do not have to be whole numbers – they can be fractional, as in the case of 16¼.This indicates that the base is raised to the power of a quarter It is more usual though torefer to this as the fourth root of the base and to write it as follows:

416

The symbol indicates a root A root is the number that produces a given number when

raised to a specified power Thus, the fourth root of 16 is 2 (2 2  2  2  16)

A square root of any number is the number which, when multiplied by itself, is equal to thefirst number Thus

29  3 (3  3  9)

The root index (the little 4 or little 2 in the above examples) can be any number It is, though,

customary to omit the 2 from the root sign when referring to the square root Thus

36 refers to the square root of 36, which is 6

Expressing Numbers in Standard Form

The exploration of indices leads us on to this very useful way of expressing numbers

Working with very large (or very small) numbers can be difficult, and it is helpful if we can find

a different way of expressing them which will make them easier to deal with Standard form

provides such a way

Consider the number 93,825,000,000,000 It is a very big number and you would probablyhave great difficulty stating it, let alone multiplying it by 843,605,000,000,000

In fact, can you even tell which is the bigger? Probably not It is made a little easier by thetwo numbers being very close to each other on the page, but if one was on a different page,

it would be very difficult indeed

However, help is at hand

The method of expressing numbers in standard form reduces the number to a value between

1 and 10, followed by the number of 10s you need to multiply it by in order to make it up tothe correct size The above two numbers, under this system are:

9.3825 1013

and 8.43605 1014

.The 13 and 14 after the 10 are index numbers, as discussed above Written out in full, thefirst number would be:

9.3825 (10  10  10  10  10  10  10  10  10  10  10  10  10)

This is not particularly useful, so an easier way of dealing with a number expressed in

standard form is to remember that the index number after the 10 tells you how many places

the decimal point has to be moved to the right in order to create the full number

(remembering also to put in the 0s) So, the second number from above, written out in fullwould be:

8 4 3 6 0 5 0 0 0 0 0 0 0 0 0 = 843,605,000,000,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Converting numbers in this way makes arithmetic operations much easier – something which

we shall return to later in your studies – and also allows us to compare numbers at a glance.Thus, we can see immediately that 8.43605 1014is larger than 9.3825 1013

Just as with indices we can also have numbers expressed in standard form using a negativeindex – for example, 2.43 102

Before we consider this in more detail, it would be helpful to recap exactly what the indexnumber means

We know that 10 10  100, so we can say that 102 100

We can go on multiplying 10 by itself to get:

A factorial is the product of all the whole numbers from a given number down to one

For example, 4 factorial is written as 4! (and is sometimes read as "four bang") and is equal

to 4 3  2  1

You may encounter factorials in certain types of statistical operations

Questions for Practice 10

1 Express the following numbers in standard form:

2 The check on your answers is achieved by adding a column into which you put the sum

of each row The total of the check column should equal the sum of the four columntotals

10

3

(d)31

(e)

7

2

(f)61

(g)

9

4

(h)31

(i)

8

1

(j)83

2 (a)

10

2

(b)2012

(c)

81

36

(d)8872

Questions for Practice 5